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997 ways (Posted on 2014-06-14) Difficulty: 3 of 5

A recent puzzle asked for the smallest number that can be written as the difference of 2 squares in 4 ways - very interesting but not that challenging.

What is the smallest number that can be written as the (positive) difference of 2 squares in exactly 997 ways?

  Submitted by broll    
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Solution: (Hide)
The first crucial insight is that for some number, a, whose divisors are {1...n,a}:

((a+1)+(a-1))((a+1)-(a-1)) = ((a/n+n)+(a/n-n))((a/n+n)-(a/n-n)) = 4a.

Let x be the smallest number of ways that some number can be written as the difference of 2 squares.

For x>3

Find the smallest number, s, with 2x divisors, or, if smaller, the smallest number with 2x+1 divisors (always a square);

List the 2x proper divisors of s {1,2,.....s/2,s}

Pair the divisors: {s,1}{s/2,2}...{s/n(x),n(x)}; if there is an odd divisor, it must be s^(1/2); ignore it, as it can't be paired.

Now for each pair of squares:

{((s+1)+(s-1))*((s+1)-(s-1)) to ((s/n(x)+ n(x))+(s/n(x)-n(x)))*(((s/n(x)+n(x))-(s/n(x)-n(x)))}

the products will be equal, hence the differences of squares will be equal too; and there will be no smaller s having this property.

Simplifying the first term, the sum of each such pair will be 4s.

Now if x=997, 2x=1994. There is a list of 2000 'Smallest number with exactly n divisors' at A005179 in Sloane. (By comparison, A094191 has only 50 terms.) The entry for x = 1994 is big:

200907863847425122677829696698750339480263402194787551/ 395703197819440822085925522967474696027942973398924001/ 172041215029722589741067308883174710829525598277573773/ 145777954330976645148844919458910211640915369621405942/ 161398763726891734353948988944929648209933956622665643/ 63305934348670725319760627630081995.

However, x = 1995 has 2x+1 divisors and must be a square: 5852528640000 = 2419200^2. Since this number is compliant with the necessary conditions, multiply by 4 to obtain 23410114560000, which will be the difference between:

(5852528640000+1)^2-(5852528640000-1)^2 and 996 other pairs of squares.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): A tiny hint NEEDEDAdy TZIDON2014-06-15 04:21:58
Hints/Tipsre: A tiny hint NEEDEDbroll2014-06-15 04:09:26
QuestionA tiny hint NEEDEDAdy TZIDON2014-06-15 03:48:36
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